Abstract

The Taylor invariant is a modification of the Fourier transform of an image. It yields a representation that uniquely characterizes the image up to position. The representation retains the rotational symmetry of the Fourier transform and is invariant under translation. The magnitude spectrum is known to exhibit the properties of invariance under translation and rotational symmetry. However, although the phase spectrum satisfies the rotational-symmetry property, it has a linear part that depends on the position of the image with respect to the planar coordinate system. The Taylor invariant can be thought of as the Fourier transform of the image with the linear part of its phase removed. The linear part of the phase can be calculated from the partial derivatives of the phase evaluated at the origin. This procedure is shown to be equivalent to taking the Fourier transform of the image represented with its center of mass (intensity) at the origin of the coordinate system. The rotational-symmetry property permits a representation that is invariant under translation, rotation, and changes of scale provided that an appropriate modification of the Fourier—Mellin transform of the Taylor invariant is made.

© 1990 Optical Society of America